LMFDB - Embedded newform 7803.2.a.cd.1.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7803,2,Mod(1,7803)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7803.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7803, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7803 = 3^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7803.a (trivial)

Newform invariants

comment:select newform

sage:traces = [24,0,0,24,-20,0,0,0,0,0,-8,0,4,-16,0,24,0,0,4,-48,0,0,-36,0, 28,0,0,0,-64,0,0,0,0,0,0,0,0,0,0,0,-36,0,4,-16,0,0,0,0,24,0,0,16,0,0,20, -80,0,0,0,0,0,16,0,-24,-72,0,16,0,0,-48,-72,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(73)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$62.3072686972$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 459)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	7803.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.533361 q^{2} -1.71553 q^{4} +2.01564 q^{5} -2.52795 q^{7} -1.98172 q^{8} +1.07506 q^{10} -2.43279 q^{11} +3.12835 q^{13} -1.34831 q^{14} +2.37408 q^{16} +5.61764 q^{19} -3.45788 q^{20} -1.29756 q^{22} -2.17884 q^{23} -0.937211 q^{25} +1.66854 q^{26} +4.33677 q^{28} -4.75129 q^{29} +4.46744 q^{31} +5.22968 q^{32} -5.09543 q^{35} +4.68703 q^{37} +2.99623 q^{38} -3.99442 q^{40} -9.39479 q^{41} -9.70418 q^{43} +4.17352 q^{44} -1.16211 q^{46} +10.4884 q^{47} -0.609457 q^{49} -0.499872 q^{50} -5.36677 q^{52} -2.83803 q^{53} -4.90362 q^{55} +5.00969 q^{56} -2.53415 q^{58} -3.50470 q^{59} -7.88955 q^{61} +2.38276 q^{62} -1.95885 q^{64} +6.30562 q^{65} -4.16861 q^{67} -2.71771 q^{70} +4.51044 q^{71} +12.5366 q^{73} +2.49988 q^{74} -9.63720 q^{76} +6.14998 q^{77} +4.52640 q^{79} +4.78528 q^{80} -5.01082 q^{82} +10.8255 q^{83} -5.17583 q^{86} +4.82111 q^{88} +16.2213 q^{89} -7.90833 q^{91} +3.73785 q^{92} +5.59411 q^{94} +11.3231 q^{95} +0.802837 q^{97} -0.325061 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{4} - 20 q^{5} - 8 q^{11} + 4 q^{13} - 16 q^{14} + 24 q^{16} + 4 q^{19} - 48 q^{20} - 36 q^{23} + 28 q^{25} - 64 q^{29} - 36 q^{41} + 4 q^{43} - 16 q^{44} + 24 q^{49} + 16 q^{52} + 20 q^{55}+ \cdots + 12 q^{95}+O(q^{100}) $ 24 * q + 24 * q^4 - 20 * q^5 - 8 * q^11 + 4 * q^13 - 16 * q^14 + 24 * q^16 + 4 * q^19 - 48 * q^20 - 36 * q^23 + 28 * q^25 - 64 * q^29 - 36 * q^41 + 4 * q^43 - 16 * q^44 + 24 * q^49 + 16 * q^52 + 20 * q^55 - 80 * q^56 + 16 * q^62 - 24 * q^64 - 72 * q^65 + 16 * q^67 - 48 * q^70 - 72 * q^71 - 80 * q^74 - 40 * q^76 - 104 * q^80 - 144 * q^92 - 72 * q^94 + 12 * q^95

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.533361	0.377143	0.188572	−	0.982059i	$-0.439614\pi$
$2$	0.533361	0.377143	0.188572	+	0.982059i	$0.439614\pi$
$3$	0	0
$3$	0	0
$4$	−1.71553	−0.857763
$5$	2.01564	0.901420	0.450710	−	0.892670i	$-0.351171\pi$
$5$	2.01564	0.901420	0.450710	+	0.892670i	$0.351171\pi$
$6$	0	0
$7$	−2.52795	−0.955476	−0.477738	−	0.878502i	$-0.658543\pi$
$7$	−2.52795	−0.955476	−0.477738	+	0.878502i	$0.658543\pi$
$8$	−1.98172	−0.700643
$9$	0	0
$10$	1.07506	0.339965
$11$	−2.43279	−0.733515	−0.366757	−	0.930317i	$-0.619532\pi$
$11$	−2.43279	−0.733515	−0.366757	+	0.930317i	$0.619532\pi$
$12$	0	0
$13$	3.12835	0.867649	0.433824	−	0.900997i	$-0.357164\pi$
$13$	3.12835	0.867649	0.433824	+	0.900997i	$0.357164\pi$
$14$	−1.34831	−0.360352
$15$	0	0
$16$	2.37408	0.593520
$17$	0	0
$17$	0	0
$18$	0	0
$19$	5.61764	1.28877	0.644387	−	0.764699i	$-0.277113\pi$
$19$	5.61764	1.28877	0.644387	+	0.764699i	$0.277113\pi$
$20$	−3.45788	−0.773205
$21$	0	0
$22$	−1.29756	−0.276640
$23$	−2.17884	−0.454319	−0.227160	−	0.973858i	$-0.572944\pi$
$23$	−2.17884	−0.454319	−0.227160	+	0.973858i	$0.572944\pi$
$24$	0	0
$25$	−0.937211	−0.187442
$26$	1.66854	0.327228
$27$	0	0
$28$	4.33677	0.819572
$29$	−4.75129	−0.882293	−0.441146	−	0.897435i	$-0.645428\pi$
$29$	−4.75129	−0.882293	−0.441146	+	0.897435i	$0.645428\pi$
$30$	0	0
$31$	4.46744	0.802375	0.401188	−	0.915996i	$-0.368598\pi$
$31$	4.46744	0.802375	0.401188	+	0.915996i	$0.368598\pi$
$32$	5.22968	0.924485
$33$	0	0
$34$	0	0
$35$	−5.09543	−0.861285
$36$	0	0
$37$	4.68703	0.770544	0.385272	−	0.922803i	$-0.374108\pi$
$37$	4.68703	0.770544	0.385272	+	0.922803i	$0.374108\pi$
$38$	2.99623	0.486053
$39$	0	0
$40$	−3.99442	−0.631573
$41$	−9.39479	−1.46722	−0.733610	−	0.679571i	$-0.762166\pi$
$41$	−9.39479	−1.46722	−0.733610	+	0.679571i	$0.762166\pi$
$42$	0	0
$43$	−9.70418	−1.47987	−0.739937	−	0.672677i	$-0.765145\pi$
$43$	−9.70418	−1.47987	−0.739937	+	0.672677i	$0.765145\pi$
$44$	4.17352	0.629182
$45$	0	0
$46$	−1.16211	−0.171344
$47$	10.4884	1.52989	0.764946	−	0.644094i	$-0.222766\pi$
$47$	10.4884	1.52989	0.764946	+	0.644094i	$0.222766\pi$
$48$	0	0
$49$	−0.609457	−0.0870652
$50$	−0.499872	−0.0706926
$51$	0	0
$52$	−5.36677	−0.744237
$53$	−2.83803	−0.389834	−0.194917	−	0.980820i	$-0.562444\pi$
$53$	−2.83803	−0.389834	−0.194917	+	0.980820i	$0.562444\pi$
$54$	0	0
$55$	−4.90362	−0.661205
$56$	5.00969	0.669448
$57$	0	0
$58$	−2.53415	−0.332751
$59$	−3.50470	−0.456273	−0.228137	−	0.973629i	$-0.573263\pi$
$59$	−3.50470	−0.456273	−0.228137	+	0.973629i	$0.573263\pi$
$60$	0	0
$61$	−7.88955	−1.01015	−0.505077	−	0.863075i	$-0.668536\pi$
$61$	−7.88955	−1.01015	−0.505077	+	0.863075i	$0.668536\pi$
$62$	2.38276	0.302610
$63$	0	0
$64$	−1.95885	−0.244857
$65$	6.30562	0.782116
$66$	0	0
$67$	−4.16861	−0.509277	−0.254639	−	0.967036i	$-0.581957\pi$
$67$	−4.16861	−0.509277	−0.254639	+	0.967036i	$0.581957\pi$
$68$	0	0
$69$	0	0
$70$	−2.71771	−0.324828
$71$	4.51044	0.535291	0.267645	−	0.963517i	$-0.413754\pi$
$71$	4.51044	0.535291	0.267645	+	0.963517i	$0.413754\pi$
$72$	0	0
$73$	12.5366	1.46730	0.733651	−	0.679526i	$-0.237815\pi$
$73$	12.5366	1.46730	0.733651	+	0.679526i	$0.237815\pi$
$74$	2.49988	0.290605
$75$	0	0
$76$	−9.63720	−1.10546
$77$	6.14998	0.700856
$78$	0	0
$79$	4.52640	0.509260	0.254630	−	0.967039i	$-0.418046\pi$
$79$	4.52640	0.509260	0.254630	+	0.967039i	$0.418046\pi$
$80$	4.78528	0.535011
$81$	0	0
$82$	−5.01082	−0.553352
$83$	10.8255	1.18825	0.594125	−	0.804373i	$-0.297499\pi$
$83$	10.8255	1.18825	0.594125	+	0.804373i	$0.297499\pi$
$84$	0	0
$85$	0	0
$86$	−5.17583	−0.558124
$87$	0	0
$88$	4.82111	0.513932
$89$	16.2213	1.71945	0.859726	−	0.510756i	$-0.170634\pi$
$89$	16.2213	1.71945	0.859726	+	0.510756i	$0.170634\pi$
$90$	0	0
$91$	−7.90833	−0.829018
$92$	3.73785	0.389698
$93$	0	0
$94$	5.59411	0.576989
$95$	11.3231	1.16173
$96$	0	0
$97$	0.802837	0.0815158	0.0407579	−	0.999169i	$-0.487023\pi$
$97$	0.802837	0.0815158	0.0407579	+	0.999169i	$0.487023\pi$
$98$	−0.325061	−0.0328361
$99$	0	0
$100$	1.60781	0.160781
$101$	−10.7867	−1.07331	−0.536657	−	0.843801i	$-0.680313\pi$
$101$	−10.7867	−1.07331	−0.536657	+	0.843801i	$0.680313\pi$
$102$	0	0
$103$	13.2358	1.30416	0.652082	−	0.758149i	$-0.273896\pi$
$103$	13.2358	1.30416	0.652082	+	0.758149i	$0.273896\pi$
$104$	−6.19951	−0.607912
$105$	0	0
$106$	−1.51370	−0.147023
$107$	8.20981	0.793673	0.396836	−	0.917889i	$-0.370108\pi$
$107$	8.20981	0.793673	0.396836	+	0.917889i	$0.370108\pi$
$108$	0	0
$109$	0.228783	0.0219135	0.0109567	−	0.999940i	$-0.496512\pi$
$109$	0.228783	0.0219135	0.0109567	+	0.999940i	$0.496512\pi$
$110$	−2.61540	−0.249369
$111$	0	0
$112$	−6.00156	−0.567094
$113$	−12.0126	−1.13005	−0.565024	−	0.825075i	$-0.691133\pi$
$113$	−12.0126	−1.13005	−0.565024	+	0.825075i	$0.691133\pi$
$114$	0	0
$115$	−4.39175	−0.409533
$116$	8.15096	0.756798
$117$	0	0
$118$	−1.86927	−0.172081
$119$	0	0
$120$	0	0
$121$	−5.08152	−0.461956
$122$	−4.20798	−0.380973
$123$	0	0
$124$	−7.66400	−0.688248
$125$	−11.9673	−1.07038
$126$	0	0
$127$	−9.51499	−0.844319	−0.422160	−	0.906522i	$-0.638728\pi$
$127$	−9.51499	−0.844319	−0.422160	+	0.906522i	$0.638728\pi$
$128$	−11.5041	−1.01683
$129$	0	0
$130$	3.36317	0.294970
$131$	−14.9234	−1.30387	−0.651934	−	0.758276i	$-0.726042\pi$
$131$	−14.9234	−1.30387	−0.651934	+	0.758276i	$0.726042\pi$
$132$	0	0
$133$	−14.2011	−1.23139
$134$	−2.22338	−0.192070
$135$	0	0
$136$	0	0
$137$	−14.1629	−1.21002	−0.605009	−	0.796219i	$-0.706830\pi$
$137$	−14.1629	−1.21002	−0.605009	+	0.796219i	$0.706830\pi$
$138$	0	0
$139$	−15.4050	−1.30664	−0.653319	−	0.757083i	$-0.726624\pi$
$139$	−15.4050	−1.30664	−0.653319	+	0.757083i	$0.726624\pi$
$140$	8.74135	0.738779
$141$	0	0
$142$	2.40569	0.201881
$143$	−7.61063	−0.636433
$144$	0	0
$145$	−9.57687	−0.795316
$146$	6.68656	0.553383
$147$	0	0
$148$	−8.04073	−0.660944
$149$	−6.62192	−0.542489	−0.271244	−	0.962511i	$-0.587435\pi$
$149$	−6.62192	−0.542489	−0.271244	+	0.962511i	$0.587435\pi$
$150$	0	0
$151$	−17.0531	−1.38776	−0.693882	−	0.720089i	$-0.744101\pi$
$151$	−17.0531	−1.38776	−0.693882	+	0.720089i	$0.744101\pi$
$152$	−11.1326	−0.902971
$153$	0	0
$154$	3.28016	0.264323
$155$	9.00472	0.723277
$156$	0	0
$157$	14.5703	1.16284	0.581418	−	0.813605i	$-0.302498\pi$
$157$	14.5703	1.16284	0.581418	+	0.813605i	$0.302498\pi$
$158$	2.41421	0.192064
$159$	0	0
$160$	10.5411	0.833349
$161$	5.50800	0.434091
$162$	0	0
$163$	5.39557	0.422614	0.211307	−	0.977420i	$-0.432228\pi$
$163$	5.39557	0.422614	0.211307	+	0.977420i	$0.432228\pi$
$164$	16.1170	1.25853
$165$	0	0
$166$	5.77388	0.448140
$167$	−12.4575	−0.963993	−0.481996	−	0.876173i	$-0.660088\pi$
$167$	−12.4575	−0.963993	−0.481996	+	0.876173i	$0.660088\pi$
$168$	0	0
$169$	−3.21341	−0.247185
$170$	0	0
$171$	0	0
$172$	16.6478	1.26938
$173$	−2.67379	−0.203284	−0.101642	−	0.994821i	$-0.532410\pi$
$173$	−2.67379	−0.203284	−0.101642	+	0.994821i	$0.532410\pi$
$174$	0	0
$175$	2.36922	0.179097
$176$	−5.77565	−0.435356
$177$	0	0
$178$	8.65180	0.648480
$179$	−14.0084	−1.04704	−0.523519	−	0.852014i	$-0.675381\pi$
$179$	−14.0084	−1.04704	−0.523519	+	0.852014i	$0.675381\pi$
$180$	0	0
$181$	−13.3634	−0.993291	−0.496646	−	0.867953i	$-0.665435\pi$
$181$	−13.3634	−0.993291	−0.496646	+	0.867953i	$0.665435\pi$
$182$	−4.21800	−0.312659
$183$	0	0
$184$	4.31784	0.318316
$185$	9.44735	0.694583
$186$	0	0
$187$	0	0
$188$	−17.9931	−1.31228
$189$	0	0
$190$	6.03931	0.438138
$191$	20.9983	1.51938	0.759691	−	0.650285i	$-0.225350\pi$
$191$	20.9983	1.51938	0.759691	+	0.650285i	$0.225350\pi$
$192$	0	0
$193$	5.11341	0.368072	0.184036	−	0.982920i	$-0.441084\pi$
$193$	5.11341	0.368072	0.184036	+	0.982920i	$0.441084\pi$
$194$	0.428202	0.0307431
$195$	0	0
$196$	1.04554	0.0746813
$197$	−3.83971	−0.273568	−0.136784	−	0.990601i	$-0.543677\pi$
$197$	−3.83971	−0.273568	−0.136784	+	0.990601i	$0.543677\pi$
$198$	0	0
$199$	−7.22725	−0.512326	−0.256163	−	0.966634i	$-0.582458\pi$
$199$	−7.22725	−0.512326	−0.256163	+	0.966634i	$0.582458\pi$
$200$	1.85729	0.131330
$201$	0	0
$202$	−5.75319	−0.404793
$203$	12.0110	0.843010
$204$	0	0
$205$	−18.9365	−1.32258
$206$	7.05947	0.491857
$207$	0	0
$208$	7.42696	0.514967
$209$	−13.6665	−0.945335
$210$	0	0
$211$	−12.5596	−0.864636	−0.432318	−	0.901721i	$-0.642304\pi$
$211$	−12.5596	−0.864636	−0.432318	+	0.901721i	$0.642304\pi$
$212$	4.86872	0.334385
$213$	0	0
$214$	4.37880	0.299328
$215$	−19.5601	−1.33399
$216$	0	0
$217$	−11.2935	−0.766650
$218$	0.122024	0.00826451
$219$	0	0
$220$	8.41230	0.567157
$221$	0	0
$222$	0	0
$223$	4.79698	0.321229	0.160615	−	0.987017i	$-0.448652\pi$
$223$	4.79698	0.321229	0.160615	+	0.987017i	$0.448652\pi$
$224$	−13.2204	−0.883323
$225$	0	0
$226$	−6.40704	−0.426190
$227$	−22.4744	−1.49168	−0.745839	−	0.666127i	$-0.767951\pi$
$227$	−22.4744	−1.49168	−0.745839	+	0.666127i	$0.767951\pi$
$228$	0	0
$229$	−2.28961	−0.151302	−0.0756510	−	0.997134i	$-0.524103\pi$
$229$	−2.28961	−0.151302	−0.0756510	+	0.997134i	$0.524103\pi$
$230$	−2.34239	−0.154452
$231$	0	0
$232$	9.41572	0.618172
$233$	−24.9870	−1.63696	−0.818478	−	0.574538i	$-0.805182\pi$
$233$	−24.9870	−1.63696	−0.818478	+	0.574538i	$0.805182\pi$
$234$	0	0
$235$	21.1408	1.37908
$236$	6.01241	0.391374
$237$	0	0
$238$	0	0
$239$	−16.8913	−1.09261	−0.546304	−	0.837587i	$-0.683966\pi$
$239$	−16.8913	−1.09261	−0.546304	+	0.837587i	$0.683966\pi$
$240$	0	0
$241$	−29.9907	−1.93187	−0.965936	−	0.258783i	$-0.916679\pi$
$241$	−29.9907	−1.93187	−0.965936	+	0.258783i	$0.916679\pi$
$242$	−2.71029	−0.174224
$243$	0	0
$244$	13.5347	0.866472
$245$	−1.22844	−0.0784823
$246$	0	0
$247$	17.5740	1.11820
$248$	−8.85320	−0.562178
$249$	0	0
$250$	−6.38287	−0.403688
$251$	−24.1719	−1.52572	−0.762858	−	0.646566i	$-0.776205\pi$
$251$	−24.1719	−1.52572	−0.762858	+	0.646566i	$0.776205\pi$
$252$	0	0
$253$	5.30066	0.333250
$254$	−5.07493	−0.318429
$255$	0	0
$256$	−2.21815	−0.138634
$257$	26.7412	1.66807	0.834034	−	0.551712i	$-0.186025\pi$
$257$	26.7412	1.66807	0.834034	+	0.551712i	$0.186025\pi$
$258$	0	0
$259$	−11.8486	−0.736236
$260$	−10.8175	−0.670870
$261$	0	0
$262$	−7.95959	−0.491745
$263$	−2.34258	−0.144449	−0.0722247	−	0.997388i	$-0.523010\pi$
$263$	−2.34258	−0.144449	−0.0722247	+	0.997388i	$0.523010\pi$
$264$	0	0
$265$	−5.72044	−0.351404
$266$	−7.57433	−0.464412
$267$	0	0
$268$	7.15136	0.436839
$269$	17.2220	1.05005	0.525023	−	0.851088i	$-0.324057\pi$
$269$	17.2220	1.05005	0.525023	+	0.851088i	$0.324057\pi$
$270$	0	0
$271$	24.8158	1.50745	0.753726	−	0.657189i	$-0.228254\pi$
$271$	24.8158	1.50745	0.753726	+	0.657189i	$0.228254\pi$
$272$	0	0
$273$	0	0
$274$	−7.55394	−0.456350
$275$	2.28004	0.137492
$276$	0	0
$277$	−18.5075	−1.11201	−0.556005	−	0.831179i	$-0.687666\pi$
$277$	−18.5075	−1.11201	−0.556005	+	0.831179i	$0.687666\pi$
$278$	−8.21645	−0.492790
$279$	0	0
$280$	10.0977	0.603453
$281$	17.2627	1.02981	0.514903	−	0.857249i	$-0.327828\pi$
$281$	17.2627	1.02981	0.514903	+	0.857249i	$0.327828\pi$
$282$	0	0
$283$	−20.7111	−1.23115	−0.615574	−	0.788079i	$-0.711076\pi$
$283$	−20.7111	−1.23115	−0.615574	+	0.788079i	$0.711076\pi$
$284$	−7.73778	−0.459153
$285$	0	0
$286$	−4.05922	−0.240027
$287$	23.7496	1.40189
$288$	0	0
$289$	0	0
$290$	−5.10793	−0.299948
$291$	0	0
$292$	−21.5069	−1.25860
$293$	−23.8401	−1.39275	−0.696375	−	0.717678i	$-0.745205\pi$
$293$	−23.8401	−1.39275	−0.696375	+	0.717678i	$0.745205\pi$
$294$	0	0
$295$	−7.06421	−0.411294
$296$	−9.28838	−0.539876
$297$	0	0
$298$	−3.53187	−0.204596
$299$	−6.81618	−0.394190
$300$	0	0
$301$	24.5317	1.41398
$302$	−9.09548	−0.523386
$303$	0	0
$304$	13.3367	0.764913
$305$	−15.9025	−0.910572
$306$	0	0
$307$	13.9729	0.797474	0.398737	−	0.917065i	$-0.369449\pi$
$307$	13.9729	0.797474	0.398737	+	0.917065i	$0.369449\pi$
$308$	−10.5505	−0.601168
$309$	0	0
$310$	4.80277	0.272779
$311$	9.75466	0.553136	0.276568	−	0.960994i	$-0.410803\pi$
$311$	9.75466	0.553136	0.276568	+	0.960994i	$0.410803\pi$
$312$	0	0
$313$	33.2415	1.87892	0.939461	−	0.342655i	$-0.111326\pi$
$313$	33.2415	1.87892	0.939461	+	0.342655i	$0.111326\pi$
$314$	7.77123	0.438556
$315$	0	0
$316$	−7.76516	−0.436825
$317$	−7.34140	−0.412334	−0.206167	−	0.978517i	$-0.566099\pi$
$317$	−7.34140	−0.412334	−0.206167	+	0.978517i	$0.566099\pi$
$318$	0	0
$319$	11.5589	0.647175
$320$	−3.94834	−0.220719
$321$	0	0
$322$	2.93775	0.163715
$323$	0	0
$324$	0	0
$325$	−2.93193	−0.162634
$326$	2.87779	0.159386
$327$	0	0
$328$	18.6178	1.02800
$329$	−26.5142	−1.46178
$330$	0	0
$331$	2.05218	0.112798	0.0563990	−	0.998408i	$-0.482038\pi$
$331$	2.05218	0.112798	0.0563990	+	0.998408i	$0.482038\pi$
$332$	−18.5714	−1.01924
$333$	0	0
$334$	−6.64436	−0.363563
$335$	−8.40241	−0.459073
$336$	0	0
$337$	−6.09966	−0.332270	−0.166135	−	0.986103i	$-0.553129\pi$
$337$	−6.09966	−0.332270	−0.166135	+	0.986103i	$0.553129\pi$
$338$	−1.71391	−0.0932243
$339$	0	0
$340$	0	0
$341$	−10.8683	−0.588554
$342$	0	0
$343$	19.2363	1.03866
$344$	19.2309	1.03686
$345$	0	0
$346$	−1.42609	−0.0766673
$347$	−34.4147	−1.84748	−0.923738	−	0.383026i	$-0.874882\pi$
$347$	−34.4147	−1.84748	−0.923738	+	0.383026i	$0.874882\pi$
$348$	0	0
$349$	−17.4294	−0.932976	−0.466488	−	0.884528i	$-0.654481\pi$
$349$	−17.4294	−0.932976	−0.466488	+	0.884528i	$0.654481\pi$
$350$	1.26365	0.0675451
$351$	0	0
$352$	−12.7227	−0.678123
$353$	−35.9886	−1.91548	−0.957739	−	0.287638i	$-0.907130\pi$
$353$	−35.9886	−1.91548	−0.957739	+	0.287638i	$0.907130\pi$
$354$	0	0
$355$	9.09141	0.482522
$356$	−27.8280	−1.47488
$357$	0	0
$358$	−7.47155	−0.394884
$359$	−18.7253	−0.988285	−0.494142	−	0.869381i	$-0.664518\pi$
$359$	−18.7253	−0.988285	−0.494142	+	0.869381i	$0.664518\pi$
$360$	0	0
$361$	12.5578	0.660939
$362$	−7.12750	−0.374613
$363$	0	0
$364$	13.5669	0.711101
$365$	25.2693	1.32266
$366$	0	0
$367$	24.2752	1.26716	0.633578	−	0.773679i	$-0.281586\pi$
$367$	24.2752	1.26716	0.633578	+	0.773679i	$0.281586\pi$
$368$	−5.17274	−0.269648
$369$	0	0
$370$	5.03885	0.261957
$371$	7.17441	0.372477
$372$	0	0
$373$	1.52714	0.0790723	0.0395361	−	0.999218i	$-0.487412\pi$
$373$	1.52714	0.0790723	0.0395361	+	0.999218i	$0.487412\pi$
$374$	0	0
$375$	0	0
$376$	−20.7851	−1.07191
$377$	−14.8637	−0.765520
$378$	0	0
$379$	−5.78502	−0.297156	−0.148578	−	0.988901i	$-0.547470\pi$
$379$	−5.78502	−0.297156	−0.148578	+	0.988901i	$0.547470\pi$
$380$	−19.4251	−0.996486
$381$	0	0
$382$	11.1997	0.573025
$383$	27.9186	1.42658	0.713288	−	0.700871i	$-0.247205\pi$
$383$	27.9186	1.42658	0.713288	+	0.700871i	$0.247205\pi$
$384$	0	0
$385$	12.3961	0.631765
$386$	2.72730	0.138816
$387$	0	0
$388$	−1.37729	−0.0699212
$389$	5.79580	0.293859	0.146929	−	0.989147i	$-0.453061\pi$
$389$	5.79580	0.293859	0.146929	+	0.989147i	$0.453061\pi$
$390$	0	0
$391$	0	0
$392$	1.20777	0.0610016
$393$	0	0
$394$	−2.04795	−0.103174
$395$	9.12358	0.459057
$396$	0	0
$397$	−25.6588	−1.28778	−0.643889	−	0.765119i	$-0.722680\pi$
$397$	−25.6588	−1.28778	−0.643889	+	0.765119i	$0.722680\pi$
$398$	−3.85474	−0.193221
$399$	0	0
$400$	−2.22501	−0.111251
$401$	1.94594	0.0971755	0.0485877	−	0.998819i	$-0.484528\pi$
$401$	1.94594	0.0971755	0.0485877	+	0.998819i	$0.484528\pi$
$402$	0	0
$403$	13.9757	0.696180
$404$	18.5048	0.920648
$405$	0	0
$406$	6.40622	0.317935
$407$	−11.4026	−0.565205
$408$	0	0
$409$	9.72007	0.480627	0.240313	−	0.970695i	$-0.422750\pi$
$409$	9.72007	0.480627	0.240313	+	0.970695i	$0.422750\pi$
$410$	−10.1000	−0.498803
$411$	0	0
$412$	−22.7064	−1.11866
$413$	8.85972	0.435958
$414$	0	0
$415$	21.8202	1.07111
$416$	16.3603	0.802129
$417$	0	0
$418$	−7.28921	−0.356527
$419$	4.92897	0.240796	0.120398	−	0.992726i	$-0.461583\pi$
$419$	4.92897	0.240796	0.120398	+	0.992726i	$0.461583\pi$
$420$	0	0
$421$	−33.8369	−1.64911	−0.824554	−	0.565783i	$-0.808574\pi$
$421$	−33.8369	−1.64911	−0.824554	+	0.565783i	$0.808574\pi$
$422$	−6.69878	−0.326092
$423$	0	0
$424$	5.62418	0.273134
$425$	0	0
$426$	0	0
$427$	19.9444	0.965177
$428$	−14.0841	−0.680783
$429$	0	0
$430$	−10.4326	−0.503104
$431$	14.9941	0.722242	0.361121	−	0.932519i	$-0.382394\pi$
$431$	14.9941	0.722242	0.361121	+	0.932519i	$0.382394\pi$
$432$	0	0
$433$	0.570805	0.0274311	0.0137156	−	0.999906i	$-0.495634\pi$
$433$	0.570805	0.0274311	0.0137156	+	0.999906i	$0.495634\pi$
$434$	−6.02350	−0.289137
$435$	0	0
$436$	−0.392483	−0.0187965
$437$	−12.2399	−0.585515
$438$	0	0
$439$	−25.4402	−1.21419	−0.607097	−	0.794628i	$-0.707666\pi$
$439$	−25.4402	−1.21419	−0.607097	+	0.794628i	$0.707666\pi$
$440$	9.71760	0.463268
$441$	0	0
$442$	0	0
$443$	−1.27194	−0.0604317	−0.0302158	−	0.999543i	$-0.509619\pi$
$443$	−1.27194	−0.0604317	−0.0302158	+	0.999543i	$0.509619\pi$
$444$	0	0
$445$	32.6962	1.54995
$446$	2.55852	0.121150
$447$	0	0
$448$	4.95189	0.233955
$449$	−30.7646	−1.45187	−0.725936	−	0.687762i	$-0.758593\pi$
$449$	−30.7646	−1.45187	−0.725936	+	0.687762i	$0.758593\pi$
$450$	0	0
$451$	22.8556	1.07623
$452$	20.6079	0.969313
$453$	0	0
$454$	−11.9870	−0.562576
$455$	−15.9403	−0.747293
$456$	0	0
$457$	23.4935	1.09898	0.549490	−	0.835500i	$-0.314822\pi$
$457$	23.4935	1.09898	0.549490	+	0.835500i	$0.314822\pi$
$458$	−1.22119	−0.0570625
$459$	0	0
$460$	7.53416	0.351282
$461$	8.51912	0.396775	0.198387	−	0.980124i	$-0.436430\pi$
$461$	8.51912	0.396775	0.198387	+	0.980124i	$0.436430\pi$
$462$	0	0
$463$	17.6689	0.821143	0.410571	−	0.911828i	$-0.365329\pi$
$463$	17.6689	0.821143	0.410571	+	0.911828i	$0.365329\pi$
$464$	−11.2799	−0.523658
$465$	0	0
$466$	−13.3271	−0.617367
$467$	−6.27728	−0.290478	−0.145239	−	0.989397i	$-0.546395\pi$
$467$	−6.27728	−0.290478	−0.145239	+	0.989397i	$0.546395\pi$
$468$	0	0
$469$	10.5381	0.486602
$470$	11.2757	0.520109
$471$	0	0
$472$	6.94533	0.319685
$473$	23.6083	1.08551
$474$	0	0
$475$	−5.26491	−0.241571
$476$	0	0
$477$	0	0
$478$	−9.00917	−0.412070
$479$	−33.8395	−1.54617	−0.773083	−	0.634304i	$-0.781287\pi$
$479$	−33.8395	−1.54617	−0.773083	+	0.634304i	$0.781287\pi$
$480$	0	0
$481$	14.6627	0.668561
$482$	−15.9959	−0.728592
$483$	0	0
$484$	8.71748	0.396249
$485$	1.61823	0.0734799
$486$	0	0
$487$	−17.3297	−0.785283	−0.392641	−	0.919692i	$-0.628439\pi$
$487$	−17.3297	−0.785283	−0.392641	+	0.919692i	$0.628439\pi$
$488$	15.6349	0.707757
$489$	0	0
$490$	−0.655204	−0.0295991
$491$	18.8939	0.852668	0.426334	−	0.904566i	$-0.359805\pi$
$491$	18.8939	0.852668	0.426334	+	0.904566i	$0.359805\pi$
$492$	0	0
$493$	0	0
$494$	9.37326	0.421723
$495$	0	0
$496$	10.6061	0.476226
$497$	−11.4022	−0.511458
$498$	0	0
$499$	−6.48069	−0.290116	−0.145058	−	0.989423i	$-0.546337\pi$
$499$	−6.48069	−0.290116	−0.145058	+	0.989423i	$0.546337\pi$
$500$	20.5301	0.918136
$501$	0	0
$502$	−12.8924	−0.575414
$503$	−16.6622	−0.742930	−0.371465	−	0.928447i	$-0.621144\pi$
$503$	−16.6622	−0.742930	−0.371465	+	0.928447i	$0.621144\pi$
$504$	0	0
$505$	−21.7420	−0.967506
$506$	2.82717	0.125683
$507$	0	0
$508$	16.3232	0.724226
$509$	−19.1265	−0.847767	−0.423884	−	0.905717i	$-0.639333\pi$
$509$	−19.1265	−0.847767	−0.423884	+	0.905717i	$0.639333\pi$
$510$	0	0
$511$	−31.6920	−1.40197
$512$	21.8252	0.964546
$513$	0	0
$514$	14.2627	0.629101
$515$	26.6786	1.17560
$516$	0	0
$517$	−25.5161	−1.12220
$518$	−6.31958	−0.277667
$519$	0	0
$520$	−12.4960	−0.547984
$521$	14.3864	0.630279	0.315139	−	0.949045i	$-0.397949\pi$
$521$	14.3864	0.630279	0.315139	+	0.949045i	$0.397949\pi$
$522$	0	0
$523$	−6.11004	−0.267173	−0.133587	−	0.991037i	$-0.542649\pi$
$523$	−6.11004	−0.267173	−0.133587	+	0.991037i	$0.542649\pi$
$524$	25.6015	1.11841
$525$	0	0
$526$	−1.24944	−0.0544781
$527$	0	0
$528$	0	0
$529$	−18.2527	−0.793594
$530$	−3.05106	−0.132530
$531$	0	0
$532$	24.3624	1.05624
$533$	−29.3902	−1.27303
$534$	0	0
$535$	16.5480	0.715432
$536$	8.26101	0.356821
$537$	0	0
$538$	9.18557	0.396018
$539$	1.48268	0.0638636
$540$	0	0
$541$	28.6602	1.23220	0.616099	−	0.787669i	$-0.288712\pi$
$541$	28.6602	1.23220	0.616099	+	0.787669i	$0.288712\pi$
$542$	13.2358	0.568526
$543$	0	0
$544$	0	0
$545$	0.461144	0.0197532
$546$	0	0
$547$	−16.9731	−0.725716	−0.362858	−	0.931844i	$-0.618199\pi$
$547$	−16.9731	−0.725716	−0.362858	+	0.931844i	$0.618199\pi$
$548$	24.2968	1.03791
$549$	0	0
$550$	1.21609	0.0518540
$551$	−26.6910	−1.13708
$552$	0	0
$553$	−11.4425	−0.486586
$554$	−9.87120	−0.419387
$555$	0	0
$556$	26.4277	1.12079
$557$	−11.2465	−0.476530	−0.238265	−	0.971200i	$-0.576579\pi$
$557$	−11.2465	−0.476530	−0.238265	+	0.971200i	$0.576579\pi$
$558$	0	0
$559$	−30.3581	−1.28401
$560$	−12.0970	−0.511190
$561$	0	0
$562$	9.20724	0.388384
$563$	1.34971	0.0568836	0.0284418	−	0.999595i	$-0.490945\pi$
$563$	1.34971	0.0568836	0.0284418	+	0.999595i	$0.490945\pi$
$564$	0	0
$565$	−24.2130	−1.01865
$566$	−11.0465	−0.464319
$567$	0	0
$568$	−8.93842	−0.375048
$569$	0.494896	0.0207471	0.0103736	−	0.999946i	$-0.496698\pi$
$569$	0.494896	0.0207471	0.0103736	+	0.999946i	$0.496698\pi$
$570$	0	0
$571$	24.9578	1.04445	0.522225	−	0.852808i	$-0.325102\pi$
$571$	24.9578	1.04445	0.522225	+	0.852808i	$0.325102\pi$
$572$	13.0562	0.545909
$573$	0	0
$574$	12.6671	0.528715
$575$	2.04203	0.0851586
$576$	0	0
$577$	−23.0850	−0.961043	−0.480521	−	0.876983i	$-0.659553\pi$
$577$	−23.0850	−0.961043	−0.480521	+	0.876983i	$0.659553\pi$
$578$	0	0
$579$	0	0
$580$	16.4294	0.682193
$581$	−27.3663	−1.13534
$582$	0	0
$583$	6.90435	0.285949
$584$	−24.8441	−1.02806
$585$	0	0
$586$	−12.7154	−0.525267
$587$	−20.9646	−0.865303	−0.432652	−	0.901561i	$-0.642422\pi$
$587$	−20.9646	−0.865303	−0.432652	+	0.901561i	$0.642422\pi$
$588$	0	0
$589$	25.0964	1.03408
$590$	−3.76777	−0.155117
$591$	0	0
$592$	11.1274	0.457333
$593$	27.5149	1.12990	0.564951	−	0.825124i	$-0.308895\pi$
$593$	27.5149	1.12990	0.564951	+	0.825124i	$0.308895\pi$
$594$	0	0
$595$	0	0
$596$	11.3601	0.465327
$597$	0	0
$598$	−3.63549	−0.148666
$599$	43.8786	1.79283	0.896415	−	0.443215i	$-0.146162\pi$
$599$	43.8786	1.79283	0.896415	+	0.443215i	$0.146162\pi$
$600$	0	0
$601$	43.3759	1.76934	0.884670	−	0.466218i	$-0.154384\pi$
$601$	43.3759	1.76934	0.884670	+	0.466218i	$0.154384\pi$
$602$	13.0843	0.533275
$603$	0	0
$604$	29.2551	1.19037
$605$	−10.2425	−0.416417
$606$	0	0
$607$	21.0232	0.853307	0.426653	−	0.904415i	$-0.359692\pi$
$607$	21.0232	0.853307	0.426653	+	0.904415i	$0.359692\pi$
$608$	29.3784	1.19145
$609$	0	0
$610$	−8.48175	−0.343416
$611$	32.8114	1.32741
$612$	0	0
$613$	30.5509	1.23394	0.616969	−	0.786987i	$-0.288360\pi$
$613$	30.5509	1.23394	0.616969	+	0.786987i	$0.288360\pi$
$614$	7.45259	0.300762
$615$	0	0
$616$	−12.1875	−0.491050
$617$	8.08373	0.325439	0.162719	−	0.986672i	$-0.447974\pi$
$617$	8.08373	0.325439	0.162719	+	0.986672i	$0.447974\pi$
$618$	0	0
$619$	−7.34432	−0.295193	−0.147597	−	0.989048i	$-0.547154\pi$
$619$	−7.34432	−0.295193	−0.147597	+	0.989048i	$0.547154\pi$
$620$	−15.4478	−0.620400
$621$	0	0
$622$	5.20276	0.208612
$623$	−41.0066	−1.64289
$624$	0	0
$625$	−19.4356	−0.777423
$626$	17.7297	0.708623
$627$	0	0
$628$	−24.9957	−0.997438
$629$	0	0
$630$	0	0
$631$	43.2402	1.72137	0.860683	−	0.509141i	$-0.170037\pi$
$631$	43.2402	1.72137	0.860683	+	0.509141i	$0.170037\pi$
$632$	−8.97005	−0.356810
$633$	0	0
$634$	−3.91562	−0.155509
$635$	−19.1788	−0.761086
$636$	0	0
$637$	−1.90660	−0.0755421
$638$	6.16507	0.244078
$639$	0	0
$640$	−23.1881	−0.916592
$641$	−9.44281	−0.372969	−0.186484	−	0.982458i	$-0.559709\pi$
$641$	−9.44281	−0.372969	−0.186484	+	0.982458i	$0.559709\pi$
$642$	0	0
$643$	−0.346171	−0.0136516	−0.00682582	−	0.999977i	$-0.502173\pi$
$643$	−0.346171	−0.0136516	−0.00682582	+	0.999977i	$0.502173\pi$
$644$	−9.44912	−0.372347
$645$	0	0
$646$	0	0
$647$	−2.21802	−0.0871996	−0.0435998	−	0.999049i	$-0.513883\pi$
$647$	−2.21802	−0.0871996	−0.0435998	+	0.999049i	$0.513883\pi$
$648$	0	0
$649$	8.52622	0.334683
$650$	−1.56378	−0.0613363
$651$	0	0
$652$	−9.25625	−0.362503
$653$	3.81435	0.149267	0.0746336	−	0.997211i	$-0.476221\pi$
$653$	3.81435	0.149267	0.0746336	+	0.997211i	$0.476221\pi$
$654$	0	0
$655$	−30.0802	−1.17533
$656$	−22.3040	−0.870824
$657$	0	0
$658$	−14.1416	−0.551299
$659$	29.3352	1.14274	0.571368	−	0.820694i	$-0.306413\pi$
$659$	29.3352	1.14274	0.571368	+	0.820694i	$0.306413\pi$
$660$	0	0
$661$	−6.32915	−0.246175	−0.123088	−	0.992396i	$-0.539280\pi$
$661$	−6.32915	−0.246175	−0.123088	+	0.992396i	$0.539280\pi$
$662$	1.09455	0.0425410
$663$	0	0
$664$	−21.4530	−0.832538
$665$	−28.6243	−1.11000
$666$	0	0
$667$	10.3523	0.400843
$668$	21.3712	0.826877
$669$	0	0
$670$	−4.48152	−0.173136
$671$	19.1936	0.740962
$672$	0	0
$673$	−27.6869	−1.06725	−0.533627	−	0.845720i	$-0.679171\pi$
$673$	−27.6869	−1.06725	−0.533627	+	0.845720i	$0.679171\pi$
$674$	−3.25332	−0.125313
$675$	0	0
$676$	5.51269	0.212026
$677$	−38.8370	−1.49263	−0.746314	−	0.665594i	$-0.768178\pi$
$677$	−38.8370	−1.49263	−0.746314	+	0.665594i	$0.768178\pi$
$678$	0	0
$679$	−2.02953	−0.0778864
$680$	0	0
$681$	0	0
$682$	−5.79675	−0.221969
$683$	−3.74165	−0.143170	−0.0715852	−	0.997434i	$-0.522806\pi$
$683$	−3.74165	−0.143170	−0.0715852	+	0.997434i	$0.522806\pi$
$684$	0	0
$685$	−28.5472	−1.09073
$686$	10.2599	0.391726
$687$	0	0
$688$	−23.0385	−0.878334
$689$	−8.87837	−0.338239
$690$	0	0
$691$	33.9745	1.29245	0.646226	−	0.763146i	$-0.276346\pi$
$691$	33.9745	1.29245	0.646226	+	0.763146i	$0.276346\pi$
$692$	4.58695	0.174370
$693$	0	0
$694$	−18.3554	−0.696763
$695$	−31.0509	−1.17783
$696$	0	0
$697$	0	0
$698$	−9.29618	−0.351866
$699$	0	0
$700$	−4.06447	−0.153622
$701$	−19.8395	−0.749327	−0.374663	−	0.927161i	$-0.622242\pi$
$701$	−19.8395	−0.749327	−0.374663	+	0.927161i	$0.622242\pi$
$702$	0	0
$703$	26.3301	0.993057
$704$	4.76548	0.179606
$705$	0	0
$706$	−19.1949	−0.722410
$707$	27.2682	1.02553
$708$	0	0
$709$	48.8784	1.83567	0.917833	−	0.396967i	$-0.129937\pi$
$709$	48.8784	1.83567	0.917833	+	0.396967i	$0.129937\pi$
$710$	4.84901	0.181980
$711$	0	0
$712$	−32.1460	−1.20472
$713$	−9.73382	−0.364535
$714$	0	0
$715$	−15.3403	−0.573694
$716$	24.0318	0.898111
$717$	0	0
$718$	−9.98736	−0.372725
$719$	−4.65960	−0.173774	−0.0868869	−	0.996218i	$-0.527692\pi$
$719$	−4.65960	−0.173774	−0.0868869	+	0.996218i	$0.527692\pi$
$720$	0	0
$721$	−33.4595	−1.24610
$722$	6.69787	0.249269
$723$	0	0
$724$	22.9252	0.852008
$725$	4.45296	0.165379
$726$	0	0
$727$	34.8089	1.29099	0.645495	−	0.763765i	$-0.276651\pi$
$727$	34.8089	1.29099	0.645495	+	0.763765i	$0.276651\pi$
$728$	15.6721	0.580846
$729$	0	0
$730$	13.4777	0.498831
$731$	0	0
$732$	0	0
$733$	−20.2207	−0.746867	−0.373434	−	0.927657i	$-0.621820\pi$
$733$	−20.2207	−0.746867	−0.373434	+	0.927657i	$0.621820\pi$
$734$	12.9475	0.477899
$735$	0	0
$736$	−11.3946	−0.420011
$737$	10.1414	0.373562
$738$	0	0
$739$	−20.7563	−0.763533	−0.381767	−	0.924259i	$-0.624684\pi$
$739$	−20.7563	−0.763533	−0.381767	+	0.924259i	$0.624684\pi$
$740$	−16.2072	−0.595788
$741$	0	0
$742$	3.82655	0.140477
$743$	14.0213	0.514391	0.257196	−	0.966359i	$-0.417202\pi$
$743$	14.0213	0.514391	0.257196	+	0.966359i	$0.417202\pi$
$744$	0	0
$745$	−13.3474	−0.489010
$746$	0.814517	0.0298216
$747$	0	0
$748$	0	0
$749$	−20.7540	−0.758335
$750$	0	0
$751$	33.7354	1.23102	0.615512	−	0.788128i	$-0.288949\pi$
$751$	33.7354	1.23102	0.615512	+	0.788128i	$0.288949\pi$
$752$	24.9003	0.908022
$753$	0	0
$754$	−7.92773	−0.288711
$755$	−34.3729	−1.25096
$756$	0	0
$757$	37.1888	1.35165	0.675824	−	0.737063i	$-0.263788\pi$
$757$	37.1888	1.35165	0.675824	+	0.737063i	$0.263788\pi$
$758$	−3.08550	−0.112071
$759$	0	0
$760$	−22.4392	−0.813956
$761$	10.1020	0.366198	0.183099	−	0.983094i	$-0.441387\pi$
$761$	10.1020	0.366198	0.183099	+	0.983094i	$0.441387\pi$
$762$	0	0
$763$	−0.578353	−0.0209378
$764$	−36.0231	−1.30327
$765$	0	0
$766$	14.8907	0.538023
$767$	−10.9639	−0.395885
$768$	0	0
$769$	−51.1798	−1.84559	−0.922796	−	0.385288i	$-0.874102\pi$
$769$	−51.1798	−1.84559	−0.922796	+	0.385288i	$0.874102\pi$
$770$	6.61162	0.238266
$771$	0	0
$772$	−8.77219	−0.315718
$773$	−21.0914	−0.758604	−0.379302	−	0.925273i	$-0.623836\pi$
$773$	−21.0914	−0.758604	−0.379302	+	0.925273i	$0.623836\pi$
$774$	0	0
$775$	−4.18693	−0.150399
$776$	−1.59100	−0.0571135
$777$	0	0
$778$	3.09125	0.110827
$779$	−52.7765	−1.89092
$780$	0	0
$781$	−10.9730	−0.392644
$782$	0	0
$783$	0	0
$784$	−1.44690	−0.0516750
$785$	29.3684	1.04820
$786$	0	0
$787$	17.8351	0.635753	0.317877	−	0.948132i	$-0.397030\pi$
$787$	17.8351	0.635753	0.317877	+	0.948132i	$0.397030\pi$
$788$	6.58713	0.234657
$789$	0	0
$790$	4.86617	0.173130
$791$	30.3672	1.07973
$792$	0	0
$793$	−24.6813	−0.876458
$794$	−13.6854	−0.485677
$795$	0	0
$796$	12.3985	0.439455
$797$	18.8627	0.668150	0.334075	−	0.942547i	$-0.391576\pi$
$797$	18.8627	0.668150	0.334075	+	0.942547i	$0.391576\pi$
$798$	0	0
$799$	0	0
$800$	−4.90131	−0.173288
$801$	0	0
$802$	1.03789	0.0366491
$803$	−30.4990	−1.07629
$804$	0	0
$805$	11.1021	0.391299
$806$	7.45410	0.262560
$807$	0	0
$808$	21.3761	0.752009
$809$	3.14126	0.110441	0.0552205	−	0.998474i	$-0.482414\pi$
$809$	3.14126	0.110441	0.0552205	+	0.998474i	$0.482414\pi$
$810$	0	0
$811$	−47.4960	−1.66781	−0.833905	−	0.551907i	$-0.813900\pi$
$811$	−47.4960	−1.66781	−0.833905	+	0.551907i	$0.813900\pi$
$812$	−20.6052	−0.723102
$813$	0	0
$814$	−6.08169	−0.213163
$815$	10.8755	0.380953
$816$	0	0
$817$	−54.5145	−1.90722
$818$	5.18431	0.181265
$819$	0	0
$820$	32.4860	1.13446
$821$	−14.7130	−0.513486	−0.256743	−	0.966480i	$-0.582649\pi$
$821$	−14.7130	−0.513486	−0.256743	+	0.966480i	$0.582649\pi$
$822$	0	0
$823$	28.3774	0.989173	0.494587	−	0.869128i	$-0.335319\pi$
$823$	28.3774	0.989173	0.494587	+	0.869128i	$0.335319\pi$
$824$	−26.2297	−0.913753
$825$	0	0
$826$	4.72543	0.164419
$827$	16.8952	0.587504	0.293752	−	0.955882i	$-0.405096\pi$
$827$	16.8952	0.587504	0.293752	+	0.955882i	$0.405096\pi$
$828$	0	0
$829$	3.03136	0.105283	0.0526417	−	0.998613i	$-0.483236\pi$
$829$	3.03136	0.105283	0.0526417	+	0.998613i	$0.483236\pi$
$830$	11.6380	0.403963
$831$	0	0
$832$	−6.12798	−0.212450
$833$	0	0
$834$	0	0
$835$	−25.1098	−0.868962
$836$	23.4453	0.810873
$837$	0	0
$838$	2.62892	0.0908146
$839$	37.4875	1.29421	0.647106	−	0.762400i	$-0.275979\pi$
$839$	37.4875	1.29421	0.647106	+	0.762400i	$0.275979\pi$
$840$	0	0
$841$	−6.42523	−0.221560
$842$	−18.0473	−0.621950
$843$	0	0
$844$	21.5462	0.741652
$845$	−6.47706	−0.222818
$846$	0	0
$847$	12.8458	0.441388
$848$	−6.73772	−0.231374
$849$	0	0
$850$	0	0
$851$	−10.2123	−0.350073
$852$	0	0
$853$	−31.7823	−1.08821	−0.544103	−	0.839018i	$-0.683130\pi$
$853$	−31.7823	−1.08821	−0.544103	+	0.839018i	$0.683130\pi$
$854$	10.6376	0.364010
$855$	0	0
$856$	−16.2695	−0.556081
$857$	−22.6117	−0.772402	−0.386201	−	0.922415i	$-0.626213\pi$
$857$	−22.6117	−0.772402	−0.386201	+	0.922415i	$0.626213\pi$
$858$	0	0
$859$	36.9481	1.26065	0.630327	−	0.776330i	$-0.282921\pi$
$859$	36.9481	1.26065	0.630327	+	0.776330i	$0.282921\pi$
$860$	33.5558	1.14424
$861$	0	0
$862$	7.99729	0.272389
$863$	−32.8262	−1.11742	−0.558709	−	0.829364i	$-0.688703\pi$
$863$	−32.8262	−1.11742	−0.558709	+	0.829364i	$0.688703\pi$
$864$	0	0
$865$	−5.38938	−0.183245
$866$	0.304445	0.0103455
$867$	0	0
$868$	19.3742	0.657604
$869$	−11.0118	−0.373550
$870$	0	0
$871$	−13.0409	−0.441874
$872$	−0.453384	−0.0153535
$873$	0	0
$874$	−6.52830	−0.220823
$875$	30.2527	1.02273
$876$	0	0
$877$	−25.2628	−0.853065	−0.426532	−	0.904472i	$-0.640265\pi$
$877$	−25.2628	−0.853065	−0.426532	+	0.904472i	$0.640265\pi$
$878$	−13.5688	−0.457925
$879$	0	0
$880$	−11.6416	−0.392438
$881$	−11.4521	−0.385832	−0.192916	−	0.981215i	$-0.561794\pi$
$881$	−11.4521	−0.385832	−0.192916	+	0.981215i	$0.561794\pi$
$882$	0	0
$883$	−37.1590	−1.25050	−0.625251	−	0.780424i	$-0.715003\pi$
$883$	−37.1590	−1.25050	−0.625251	+	0.780424i	$0.715003\pi$
$884$	0	0
$885$	0	0
$886$	−0.678403	−0.0227914
$887$	20.5460	0.689868	0.344934	−	0.938627i	$-0.387901\pi$
$887$	20.5460	0.689868	0.344934	+	0.938627i	$0.387901\pi$
$888$	0	0
$889$	24.0535	0.806727
$890$	17.4389	0.584552
$891$	0	0
$892$	−8.22934	−0.275539
$893$	58.9201	1.97169
$894$	0	0
$895$	−28.2359	−0.943821
$896$	29.0819	0.971558
$897$	0	0
$898$	−16.4087	−0.547564
$899$	−21.2261	−0.707930
$900$	0	0
$901$	0	0
$902$	12.1903	0.405892
$903$	0	0
$904$	23.8055	0.791760
$905$	−26.9357	−0.895372
$906$	0	0
$907$	−32.9972	−1.09565	−0.547827	−	0.836592i	$-0.684545\pi$
$907$	−32.9972	−1.09565	−0.547827	+	0.836592i	$0.684545\pi$
$908$	38.5554	1.27951
$909$	0	0
$910$	−8.50194	−0.281837
$911$	6.63185	0.219723	0.109861	−	0.993947i	$-0.464959\pi$
$911$	6.63185	0.219723	0.109861	+	0.993947i	$0.464959\pi$
$912$	0	0
$913$	−26.3361	−0.871598
$914$	12.5305	0.414473
$915$	0	0
$916$	3.92789	0.129781
$917$	37.7257	1.24581
$918$	0	0
$919$	58.7318	1.93738	0.968692	−	0.248265i	$-0.0798604\pi$
$919$	58.7318	1.93738	0.968692	+	0.248265i	$0.0798604\pi$
$920$	8.70320	0.286936
$921$	0	0
$922$	4.54377	0.149641
$923$	14.1103	0.464445
$924$	0	0
$925$	−4.39274	−0.144432
$926$	9.42390	0.309689
$927$	0	0
$928$	−24.8477	−0.815666
$929$	−49.7859	−1.63342	−0.816711	−	0.577047i	$-0.804205\pi$
$929$	−49.7859	−1.63342	−0.816711	+	0.577047i	$0.804205\pi$
$930$	0	0
$931$	−3.42371	−0.112207
$932$	42.8659	1.40412
$933$	0	0
$934$	−3.34806	−0.109552
$935$	0	0
$936$	0	0
$937$	−22.6222	−0.739035	−0.369517	−	0.929224i	$-0.620477\pi$
$937$	−22.6222	−0.739035	−0.369517	+	0.929224i	$0.620477\pi$
$938$	5.62059	0.183519
$939$	0	0
$940$	−36.2676	−1.18292
$941$	33.9667	1.10728	0.553642	−	0.832755i	$-0.313238\pi$
$941$	33.9667	1.10728	0.553642	+	0.832755i	$0.313238\pi$
$942$	0	0
$943$	20.4697	0.666586
$944$	−8.32045	−0.270807
$945$	0	0
$946$	12.5917	0.409392
$947$	−11.6328	−0.378016	−0.189008	−	0.981976i	$-0.560527\pi$
$947$	−11.6328	−0.378016	−0.189008	+	0.981976i	$0.560527\pi$
$948$	0	0
$949$	39.2190	1.27310
$950$	−2.80810	−0.0911068
$951$	0	0
$952$	0	0
$953$	−57.8467	−1.87384	−0.936919	−	0.349547i	$-0.886336\pi$
$953$	−57.8467	−1.87384	−0.936919	+	0.349547i	$0.886336\pi$
$954$	0	0
$955$	42.3249	1.36960
$956$	28.9775	0.937199
$957$	0	0
$958$	−18.0487	−0.583127
$959$	35.8031	1.15614
$960$	0	0
$961$	−11.0420	−0.356194
$962$	7.82051	0.252143
$963$	0	0
$964$	51.4498	1.65709
$965$	10.3068	0.331787
$966$	0	0
$967$	27.3184	0.878501	0.439250	−	0.898365i	$-0.355244\pi$
$967$	27.3184	0.878501	0.439250	+	0.898365i	$0.355244\pi$
$968$	10.0701	0.323666
$969$	0	0
$970$	0.863100	0.0277125
$971$	10.0632	0.322943	0.161471	−	0.986877i	$-0.448376\pi$
$971$	10.0632	0.322943	0.161471	+	0.986877i	$0.448376\pi$
$972$	0	0
$973$	38.9432	1.24846
$974$	−9.24298	−0.296164
$975$	0	0
$976$	−18.7304	−0.599546
$977$	−26.5667	−0.849945	−0.424973	−	0.905206i	$-0.639716\pi$
$977$	−26.5667	−0.849945	−0.424973	+	0.905206i	$0.639716\pi$
$978$	0	0
$979$	−39.4630	−1.26124
$980$	2.10743	0.0673192
$981$	0	0
$982$	10.0773	0.321578
$983$	−30.5443	−0.974213	−0.487106	−	0.873343i	$-0.661948\pi$
$983$	−30.5443	−0.974213	−0.487106	+	0.873343i	$0.661948\pi$
$984$	0	0
$985$	−7.73947	−0.246600
$986$	0	0
$987$	0	0
$988$	−30.1486	−0.959154
$989$	21.1438	0.672335
$990$	0	0
$991$	−2.06918	−0.0657298	−0.0328649	−	0.999460i	$-0.510463\pi$
$991$	−2.06918	−0.0657298	−0.0328649	+	0.999460i	$0.510463\pi$
$992$	23.3632	0.741784
$993$	0	0
$994$	−6.08148	−0.192893
$995$	−14.5675	−0.461821
$996$	0	0
$997$	29.7880	0.943395	0.471697	−	0.881761i	$-0.343642\pi$
$997$	29.7880	0.943395	0.471697	+	0.881761i	$0.343642\pi$
$998$	−3.45655	−0.109415
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7803.2.a.cd.1.15		24
3.2	odd	2		7803.2.a.cg.1.10		24
17.10	odd	16		459.2.l.a.406.5	✓	48
17.12	odd	16		459.2.l.a.433.5	yes	48
17.16	even	2		7803.2.a.cg.1.15		24
51.29	even	16		459.2.l.a.433.8	yes	48
51.44	even	16		459.2.l.a.406.8	yes	48
51.50	odd	2	inner	7803.2.a.cd.1.10		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.2.l.a.406.5	✓	48	17.10	odd	16
459.2.l.a.406.8	yes	48	51.44	even	16
459.2.l.a.433.5	yes	48	17.12	odd	16
459.2.l.a.433.8	yes	48	51.29	even	16
7803.2.a.cd.1.10		24	51.50	odd	2	inner
7803.2.a.cd.1.15		24	1.1	even	1	trivial
7803.2.a.cg.1.10		24	3.2	odd	2
7803.2.a.cg.1.15		24	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7803 = 3^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7803.a (trivial)

\(f(q)\)	\(=\)	\(q+0.533361 q^{2} -1.71553 q^{4} +2.01564 q^{5} -2.52795 q^{7} -1.98172 q^{8} +1.07506 q^{10} -2.43279 q^{11} +3.12835 q^{13} -1.34831 q^{14} +2.37408 q^{16} +5.61764 q^{19} -3.45788 q^{20} -1.29756 q^{22} -2.17884 q^{23} -0.937211 q^{25} +1.66854 q^{26} +4.33677 q^{28} -4.75129 q^{29} +4.46744 q^{31} +5.22968 q^{32} -5.09543 q^{35} +4.68703 q^{37} +2.99623 q^{38} -3.99442 q^{40} -9.39479 q^{41} -9.70418 q^{43} +4.17352 q^{44} -1.16211 q^{46} +10.4884 q^{47} -0.609457 q^{49} -0.499872 q^{50} -5.36677 q^{52} -2.83803 q^{53} -4.90362 q^{55} +5.00969 q^{56} -2.53415 q^{58} -3.50470 q^{59} -7.88955 q^{61} +2.38276 q^{62} -1.95885 q^{64} +6.30562 q^{65} -4.16861 q^{67} -2.71771 q^{70} +4.51044 q^{71} +12.5366 q^{73} +2.49988 q^{74} -9.63720 q^{76} +6.14998 q^{77} +4.52640 q^{79} +4.78528 q^{80} -5.01082 q^{82} +10.8255 q^{83} -5.17583 q^{86} +4.82111 q^{88} +16.2213 q^{89} -7.90833 q^{91} +3.73785 q^{92} +5.59411 q^{94} +11.3231 q^{95} +0.802837 q^{97} -0.325061 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{4} - 20 q^{5} - 8 q^{11} + 4 q^{13} - 16 q^{14} + 24 q^{16} + 4 q^{19} - 48 q^{20} - 36 q^{23} + 28 q^{25} - 64 q^{29} - 36 q^{41} + 4 q^{43} - 16 q^{44} + 24 q^{49} + 16 q^{52} + 20 q^{55}+ \cdots + 12 q^{95}+O(q^{100}) \) 24 * q + 24 * q^4 - 20 * q^5 - 8 * q^11 + 4 * q^13 - 16 * q^14 + 24 * q^16 + 4 * q^19 - 48 * q^20 - 36 * q^23 + 28 * q^25 - 64 * q^29 - 36 * q^41 + 4 * q^43 - 16 * q^44 + 24 * q^49 + 16 * q^52 + 20 * q^55 - 80 * q^56 + 16 * q^62 - 24 * q^64 - 72 * q^65 + 16 * q^67 - 48 * q^70 - 72 * q^71 - 80 * q^74 - 40 * q^76 - 104 * q^80 - 144 * q^92 - 72 * q^94 + 12 * q^95

Introduction

L-functions

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